https://www.quantamagazine.org/without-a...-20181031/
INTRO: Four researchers have recently come out with a model that upends the conventional wisdom in their field. They have used intensive computational data to suggest that for decades, if not longer, prevailing opinion about a fundamental concept has been wrong.
These are not biologists, climatologists or physicists. They don’t come from a field in which empirical models get a say in determining what counts as true. Instead they are mathematicians, representatives of a discipline whose standard currency — indisputable logical proof — normally spares them the kinds of debates that consume other fields. Yet here they are, model in hand, suggesting that it might be time to re-evaluate some long-held beliefs.
The model, which was
posted online in 2016 and is forthcoming in the
Journal of the European Mathematical Society, concerns a venerable mathematical concept known as the “rank” of an algebraic equation. The rank is a measurement that tells you something about how many of the solutions to that equation are rational numbers as opposed to irrational numbers. Equations with higher ranks have larger and more complicated sets of rational solutions.
Since the early 20th century mathematicians have wondered whether there is a limit to how high the rank can be. At first almost everyone thought there had to be a limit. But by the 1970s the prevailing view had shifted — most mathematicians had come to believe that rank was unbounded, meaning it should be possible to find curves with infinitely high ranks. And that’s where opinion stuck even though, in the eyes of some mathematicians, there weren’t any strong arguments in support of it.
“It was very authoritarian the way people said it was unbounded. But when you looked into it, the evidence seemed very slim,” said
Andrew Granville, a mathematician at the University of Montreal and University College London.
Now evidence points in the opposite direction. In the two years since the model was released, it has convinced many mathematicians that the rank of a specific type of algebraic equation really is bounded. But not everyone finds the model persuasive. The lack of resolution raises the kinds of questions that don’t often attend mathematical results — what weight should you give to empirical evidence in a field where all that really counts is proof?
“There is really no mathematical justification for why this model is exactly what we want,” said
Jennifer Park, a mathematician at Ohio State University and a co-author of the work. “Except that experimentally, a lot of things seem to be working out.”
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